How to intersect objects and delete leftover internal planes?
Select the objects and hit Ctrl+J to join them into one object, change to edit mode and select the meshes you want to intersect, hit Ctrl+F and select Intersect (Knife) from the menu and change to Self Intersect in operator panel.



After the operation is done, intersecting edges will be selected and you can mark them as seams Ctrl+E>Mark Seam, then select internal faces using L button when hovering with the mouse cursor over internal faces while in face selection mode.